What is the proper way of importing rigs from one blend file to another?
You'd better select in the source file all elements you need, Ctrl G create a group, rename it and then save, close, open the new file and APPEND that group. If you use Link instead of append, your rig will not be directly editable in your new file, you will have to make the edits in the source file.
To avoid confusion create a group even with a single element (the rig you need), then you can delete it in the new file: everything will automatically imported correctly.
Anyway, if you need only one rig (one armature with all related bones and constraints) you can rename it and simply append that armature.